This is an example of a false proof. This proof may seem sound when accompanied by an innacurate drawing. But the above picture clearly shows the error of the proof.
Proof: All triangles are isosceles
M is the midpoint of AB. P is the intersection of the perpendicular of AB and the angular bisector of angleBAC. The green lines are the perpediculars of AB and AC.
BM = MC
angleBMP = anglePMC = pi/2
angleBAP = anglePAC
triangleANP = triangleAQP
The error in this proof is in the assumption that the bisector and the perpendicular will always meet at a point interior to the triangle. As can be seen here, the point P is outside the triangle, even when the points are moved to make different triangles.
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